Multivariate Calculus - Bruce E. Shapiro

Lecture 16
Double Integrals over
Rectangles
Recall how we defined the Riemann integral as an area in Calculus I: to find the area
under a curve from a to b we partitioned the interval [a, b] into the set of numbers
a = x 0 < x 1 < x 2 < · · · < x n−1 < x n = b
We then chose a sequence of points, one inside each interval c i ∈ (x i−1 , x i ),
x i−1 < c i < x i
and approximated the area A i under the curve from x i−1 to x i with a rectangle with
a base width of δ i = x i − x i−1 and height f(c i ), so that
A i = δ i f(c i )
Figure 16.1: Calculation of the Riemann Sum to find the area under the curve from
a = x 1 to b = x n approximates the area by a sequence of rectangles and then takes
the limit as the number of rectangles becomes large.
The total area under the curve from a to b was then approximated by a sequence
of such rectangles, as the Riemann Sum
A ≈
n∑
A i =
i=1
n∑
δ i f(c i )
i=1
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